Before we draw conclusions from this result, we must investigate the meaning of a cross-ratio somewhat more fully.
If we take the points in a different order, the value of the cross-ratio will change.
The cross-ratio of four points equals that of the four conjugate points.
If this cross-ratio equals −1 the four points are said to be four harmonic points.
It is convenient to let the point at infinity occupy the last place in the symbolic expression for the cross-ratio.
If this cross-ratio equals −1 the four tangents are said to be four harmonic tangents.
We know that the cross-ratio of four points is equal to that of the corresponding row.
The cross-ratio of any four points in one equals that of the corresponding points in the other.